Shintani-domain formula for the Brumer–Stark unit

Let $e$ be the order of $\mathfrak p$ in $G_{\mathfrak f}$, and suppose that $\mathfrak p^e=(\pi)$ with $\pi$ totally positive and $\pi\equiv1\pmod{\mathfrak f}$. Let $\mathcal D$ be a Shintani domain, let $\lambda$ be $\pi$-good for $\mathcal D$, and let $\mathfrak b$ be a fractional ideal of $F$ relatively prime to $S$ and $\overline\lambda$. Shintani-domain formula conjecture. The element $u_{\mathfrak p,\lambda}(\mathfrak b,\mathcal D)\in F_{\mathfrak p}^*$ depends only on the class of $\mathfrak b\in G_{\mathfrak f}/\langle\mathfrak p\rangle$ and no other choices, including $\mathcal D$, so it may be written $u_{\mathfrak p,\lambda}(\sigma_{\mathfrak b})$. It lies in $\mathcal U_{\mathfrak p}$ and satisfies $u_{\mathfrak p,\lambda}(\sigma_{\mathfrak b})\equiv1\pmod\lambda$. Moreover, for every fractional ideal $\mathfrak a$ of $F$ prime to $S$ and $\overline\lambda$,

$$u_{\mathfrak p,\lambda}(\sigma_{\mathfrak a\mathfrak b})=u_{\mathfrak p,\lambda}(\sigma_{\mathfrak b})^{\sigma_{\mathfrak a}}.$$

This is the first author's conjectural $p$-adic analytic formula. The source gives no resolution evidence for this candidate, although the present paper proves equality of the resulting formula with the Brumer–Stark element.